Recovery Before Redemption: A Theory of Delays in Sovereign Debt Renegotiations

Transcription

1 March 22, 2009 Recovery Before Redemption: A Theory of Delays in Sovereign Debt Renegotiations David Benjamin State University of New York, Buffalo Mark L. J. Wright University of California, Los Angeles ABSTRACT Negotiations to restructure sovereign debts are protracted, taking on average 8 years to complete. In this paper we construct a new database (the most extensive of its kind covering ninety recent sovereign defaults) and use it to document that these negotiations are also ineffective in both repaying creditors and reducing the debt burden countries face. Specifically, we find that average creditor losses exceed 40 per-cent, and that the average debtor exits default with debt levels 25 per-cent higher (scaled by GDP) than when they entered default. To explain this apparent large inefficiency in negotiations, we present a theory of sovereign debt renegotiation in which delay arises from the same commitment problems that lead to default in the first place. A debt restructuring generates surplus for the parties at both the time of settlement and in the future. However, a creditor s ability to share in the future surplus is limited by the risk that the debtor will default on the settlement agreement. Hence, the debtor and creditor find it privately optimal to delay restructuring until future default risk is low, even though delay means some gains from trade remain unexploited. We show that a quantitative version of our theory can account for a number of stylized facts about sovereign default, as well as the new facts about debt restructurings that we document in this paper. Finally, we argue that our findings shed light on the existence of delays in bargaining in a wider range of contexts. We thank Lee Ohanian, Christoph Trebesch, Vivian Yue, numerous seminar participants, as well as participants at the 2007 Society for Economic Dynamics meetings in Prague, for comments. Further comments welcome. Benjamin: db64@buffalo.edu. Wright: mlwright@econ.ucla.edu.

2 1 Introduction In many economic environments, agents appear to have trouble reaching mutually advantageous agreements. In this paper, we document that this phenomenon is especially severe in the case of debt restructuring negotiations between a sovereign country in default and its international creditors. Using a new database of sovereign debt restructuring outcomes, the most extensive of its kind covering ninety recent sovereign defaults, we show that the average default takes more than 8 years to resolve, results in creditor losses (or haircuts ) of more than 40 per-cent, and leaves the sovereign country with a debt level (scaled by GDP) that is 25 per-cent higher than when they entered default. To explain this apparent inefficiency, we present a theory of sovereign borrowing, default, and debt restructuring in which delays in debt restructuring are the result of the same commitment problems that lead to default in the first place. As a debt restructuring agreement produces gains for the debtor country both in the period of the settlement, and in the future, the country would like to promise a share of these future gains as part of a settlement. However, there is a risk that the country will default on such a promise. As a result, both the country and its creditors find it privately optimal to delay restructuring until future default risk is low. We show that a quantitative version of the theory can account for a number of stylized facts about sovereign default, as well as the new facts on debt restructurings that we document in this paper. Finally, we use the theory to examine the efficacy of bailouts by multilateral institutions as a tool for both providing insurance to debtor countries, and for encouraging a prompt restructuring. We begin by presenting our database of sovereign debt restructuring outcomes. Drawn from a variety of sources, the database covers 90 defaults by 73 countries that were settled during the period 1989 to 2006, and contains data on the occurrence of default and settlement, the outcomes of negotiations, as well as measures of economic performance and indebtedness. In addition to the three facts introduced above, we emphasize three facts about the relationship of these outcomes to economic activity, and to each other, that motivate the development of our theory below. Specifically, we find that longer defaults are correlated with larger haircuts, and that there is a modest (but only a modest) tendency for countries to enter default when output is relatively low, and to emerge from default once output has recovered to its trend. Finally, we also establish that longer defaults and larger haircuts are more likely when

3 economic conditions in the defaulting country are weak at the time of default. We then present our theory of sovereign borrowing, default and debt restructuring. In the theory, international debt markets are incomplete so that default offers the sovereign country partial (and costly) insurance against adverse economic outcomes. While in default, and until it has settled with its creditors, output in the country is reduced, and access to international financial markets is limited. As a result, surplus is wasted while the country is in default, and the country and its creditors bargain over shares of this surplus. Bargaining takes place under complete information, with the bargaining power of the parties fluctuating over time. A settlement consists of a transfer of current resources as well as a new debt issue which serves to share the future surplus generated by a settlement. The value of a settlement to creditors, therefore, depends on the market value of the new debt issue, which is in turn limited by the fact that the country may default on these debts. Delay arises as both the country and creditor find it optimal to wait until the value of any debt issued as part of a settlement has recovered before agreeing on a settlement and redeeming the old debts. We next take the theory to the data and show that it is capable of matching the new debt restructuring facts above, as well as a number of facts about sovereign borrowing and default stressed in previous studies. Calibrating bargaining power in our model to the relationship between default and economic activity in the data, we generate some defaults when output is high as a result of a favorable bargaining position for the debtor. Other defaults occur following a sequence of low income levels. In such cases, the possibility of a settlement leads creditors to lend even when default risk is high, supporting higher levels of borrowing (at face value) at higher interest rates than in previous models. Defaults occur when the ability to raise debt levels in response to another negative income shock is limited. When debt levels are high, settlements consist largely of new debt issues, and occur only after significant improvements in economic circumstances or bargaining conditions that raise the value of new debt issues. This is the source of delay in our model. Likewise, when the face value of the defaulted debt is high we get large haircuts, generating a positive correlation between delay and haircuts. Since countries exit default when circumstances have improved, they are able to borrow more than they could just prior to default. Thus, debt levels often rise upon exit from default. The volatility of sovereign spreads is increased by both volatility 2

4 in the size of the expected settlement, and the greater variability in debt levels. Our paper contributes to a number of literatures. We believe we are the first to characterize the empirical relationship between delay, haircuts and debt levels for sovereign countries in default, while our characterization of creditor losses for ninety defaults triples the number of estimates previously available (e.g. Cline 1995 and Sturzenegger and Zettelmeyer 2007). Our theory contributes to the recent literature on debt and default in both an international (Arellano 2007, Kovrijnykh and Szentes 2007, Yue 2007, and Mendoza and Yue 2008) and domestic (Chatterjee et al 2007) context. Unlike all of these papers, our theory generates delays in bargaining, and does so without appealing to collective action problems among creditors (unlike Pitchford and Wright 2007, 2008), and while simultaneously explaining the evolution of debt during the default restructuring process (unlike Bi 2008 and d Erasmo 2008). Finally, we view our work as a contribution to the broader literature on delays in bargaining. Our finding that delays are predictable leads us to focus on commitment problems with complete information, and abstract from the role of asymmetric information (unlike the work surveyed by Ausubel, Cramton, and Deneckere 2002). Our approach extends the abstract bargaining environment of Merlo and Wilson (1995) by allowing for outside options, flow payoffs, and an endogenous terminal payoff. The rest of this paper is organized as follows. Section 2 describes our database of sovereign debt restructuring outcomes and presents our empirical findings. Section 3 presents our theory, first analyzing the debt restructuring process taking borrowing outcomes as given, before analyzing borrowing outcomes taking the debt restructuring process as given. We then combine the restructuring environment with the borrowing environment and provide a proof of existence of an equilibrium for the overall model. Section 4 shows that a calibrated version of the model can match the facts introduced in Section 2, while Section 5 evaluates the effect of supranational bailout policies. Section 6 concludes by reinterpreting the phenomenon of worldwide sovereign debt crises in the light of our results, and considering the theories implications for negotiations in other contexts. An appendix collects tables and figures, proofs of theorems, and provides further details on our database. 3

5 2 Sovereign Debt Restructuring Facts In this section we describe our database of sovereign defaults and debt renegotiation outcomes, and present our empirical findings. 2.A Data Sources and Construction In setting the limits of our database, we restrict attention to defaults on sovereign debts owed to private sector creditors, like banks and bondholders. The reason is that, in our model of debt restructuring below, creditors bargain with a view to maximizing the value of their settlement, and official creditors like the International Monetary Fund and creditor country governments are arguably motivated by broader concerns of equity. We define sovereign debts to include debts owed either directly by a country s national government, or owed indirectly by virtue of a government guarantee. The most comprehensive and widely used source of data on the dates of defaults on sovereign debts owed to private sector creditors, as well as the dates of settlements of these defaults, is published by the ratings agency Standard and Poors (Beers and Chambers 2006). Standard and Poors (S&P) defines a default on a debt contract to have occurred if a payment is not made within any grace period specified in the contract, or if debts are rescheduled on terms less favorable than those specified in the original debt contract. S&P defines the end of a default as occurring when a settlement occurs, typically in the form of an exchange of new debt for old debt, and when they conclude that no further near-term resolution of creditors claims is likely (page 22). Defining a default to have begun when debts are rescheduled on unfavorable terms, which is also related to the definition of a settlement, may result in an underestimate of actual delays in bargaining. Standard and Poors record only the year in which a default started and ended, and so we supplement these dates with data from Arteta and Hale (2007), Pitchford and Wright (2007) and Trebesch (2008), as well as a range of primary sources, to come up with the month, and in some cases the day, in which a default started and ended. The most novel part of our dataset lies in its estimates of creditor losses, or haircuts, for a large number of defaults. Until now, there has existed only a small number of estimates produced by different researchers using different methods for largely non-overlapping sam- 4

6 ples of defaults 1. In order to obtain the largest sample possible, and to ensure consistency of treatment, we base our measures on the World Bank s estimates of debt stock reduction, interest and principal forgiven, and debt buybacks, as published in Global Development Finance (GDF). We combine the World Bank s estimates of the reduction in the face value of the debt with estimates of the forgiveness of arrears on interest and principle. As the World Bank data do not make any distinction between forgiveness of debts by private creditors and forgiveness by official creditors, we scale the amount of forgiveness using estimates of the total amount of debt renegotiated, and on the proportion owed to private creditors, from both GDF and Institute for International Finance (2001). Losses in different years were added together and discounted back to the time of the default using a ten per-cent discount rate, following the practice of the OECD Development Assistance Committee. As shown in Appendix C, our estimates correlate closely with those of other studies. The resulting series on private creditor haircuts covers ninety defaults and renegotiations by seventy-three separate countries that were completed after GDF data on debt forgiveness first became available in 1989 and that ended prior to Our data on default dates and haircuts were then combined with data on various indicators of economic activity taken from the World Bank s World Development Indicators publication, and with data on the stock of long term sovereign debt outstanding from GDF. Short term debt is not included because it is not available disaggregated by type of creditor. 2.B The Facts Table 1 presents some summary statistics on the length of time taken to settle a default, which we refer to as delay, and on average haircuts weighted by the level of outstanding debt. There are three instances of defaults being contiguous in time, in the sense that S&P dates a default by a country as ending in the same year, or year before, another default begins 2. We present results treating these defaults both as separate events ( delay 1 ), and treating them 1 We have uncovered estimates of haircuts in 27 defaults, constructed by four different authors using five different methods. All of the estimates are tabulated for the purposes of comparison in Appendix C. 2 The three episodes are: Ecuador, who S&P treat as being in default from 1999 to 2000, and again from 2000 to 2001; Russia, in default from 1991 to 1997, and from 1998 to 2000; and Venezuela, in default from 1995 to 1997, and in

7 as a single default episode ( delay 2 ), although for the reasons raised above, we focus on the latter. Treating contiguous defaults as distinct defaults, there are ninety defaults in our sample lasting an average of 7.5 years. Delays rise to an average of 8.1 years if contiguous defaults are treated as a single default event. This finding is consistent with those of other authors, such as Pitchford and Wright (2008), who find that defaults took an average of 8.8 years to settle over the entire period from the end of the Napoleonic Wars to the present. This leads to our first result: Fact 1: sovereign defaults are time consuming to resolve, taking more than eight years on average in our sample. Table 1 also presents evidence on the average size of haircuts, where the average is weighted by the value of outstanding debts for the case of contiguous defaults. As shown in the Table, the average creditor group experienced a haircut of 44 per-cent of the value of the debt. Further information on the sizes of haircuts and delays is presented in Figure 1 which contains a scatter plot of haircuts and delays for each of the ninety settlements contained in our sample. As shown in the Figure, haircuts in our sample have ranged from approximately zero all the way up to ninety per-cent of the value of creditors claims in the case of some African defaults. Likewise, there is a great deal of variation in delays with many defaults being settled almost immediately while others are settled in excess of two decades. There is also a noticeable positive relationship between the amount of delay in renegotiation and the size of the haircut, with the correlation coefficient between the two series equalling This gives rise to our next two results: Fact 2: creditor losses (or haircuts) are substantial, with the average creditor experiencing a reduction in the value of their claim of forty-four per-cent. Fact 3: longer defaults are associated with larger haircuts, with a correlation between the length of the renegotiation process and the size of the creditor haircut of two-thirds. One possible explanation for Fact 3 is that there is a common factor driving both longer defaults and larger haircuts. To examine this, Table 1 also presents evidence on the relationship between delays and haircuts and the level of economic activity in the year of the 6

8 default. In particular, the third column shows that the larger is the decline in output in the year of default, the longer the delay and the larger the haircut, on average. The relationship is only modest, however, never rising above 0.3 in absolute value, with the correlation to haircuts barely different from zero. The fourth column Table 1 presents the relationship between delays and haircuts and the growth of output in the two years surrounding the default and finds a stronger negative relationship with haircuts. fact: This leads to our fourth Fact 4: larger output declines in the year of default are associated with modestly longer defaults and larger haircuts, with correlation coefficients around 0.25 Table 2 provides further evidence on the relationship between defaults, settlements and output. As shown in the first column, there is a broad tendency for default to be associated with adverse economic conditions, with a mean level of output roughly one-half of one percent below trend 3, while output in non-default periods is above trend by an equal amount on average. Economic adversity is particularly likely in the first year of a default, when output was on average 1.3 per-cent below trend, and tends to have dissipated by the time a country settles with its creditors when output is on average only 0.2% below trend. Nonetheless, there is a great deal of variation across country experiences so that the overall relationship between output and default is quite weak. In almost one-third of cases, a country defaults with output above trend. This confirms the earlier finding of Tomz and Wright (2007) for a larger sample of defaults, and leads to our fifth result: Fact 5: defaults are somewhat more likely to occur when output is below trend, and settlements tend to occur when output has returned to trend, with 64% of defaults beginning when output is below trend, and 49% ending when output is above trend. The average deviation of output from trend is 1.3% in the first year of a default, and 0.2% in the year of the settlement. 3 Deviations from trend are calculated using a Hodrick-Prescott filter with smoothing parameter 6.25 for annual data (see the discussion in Ravn and Uhlig 2002). Tomz and Wright (2007) establish that these facts are robust to different filtering methods. 7

9 Table 2 also explores the relationship between defaults and debt levels for the defaulting country. As shown in the table, being in default is associated with levels of debt to GDP that are more than seventy per-cent higher than for when a country is not in default, bearing in mind that our sample of countries is conditioned upon having defaulted once during this period. Strikingly, the table reveals that countries tend to exit default with levels of debt that are 25 per-cent higher than they possessed when they entered default. This figure is accentuated by some outlier countries, but even the median country exits default with 10 per-cent more debt. From this we conclude that renegotiations are ineffective at reducing the indebtedness of a debtor country. This leads to our sixth result: Fact 6: default resolution is associated with increased country indebtedness, with the average country exiting default with a debt to GDP ratio 25 per-cent higher than before they entered default. Finally, Table 1 also shows that delays and haircuts are essentially unrelated to the initial level of indebtedness of a country. In our theory, which we begin to outline in the next section, we therefore do not focus upon differences in debt levels as a major factor in negotiations. 3 A Theory of Sovereign Debt, Default, and Debt Restructuring In this section, we present our theory of sovereign borrowing and default. We begin by first describing the decisions facing a sovereign country that is in good standing with its creditors, before moving on to a description of international credit markets, and then to the debt restructuring environment, devoting the most detail to the latter. 3.A The Borrowing and Default Environment The Sovereign Borrower Consider a world in which time is discrete and lasts forever. In each period t = 0, 1,.., a sovereign country receives an endowment of the single non-storeable consumption good e (s) that is a function of the exogenous state s which takes on values in the finite set S. Thus, the endowment also takes on only a finite number, N e, of values. The state s summarizes all sources of uncertainty in the model and evolves according to a first order Markov process with transition probabilities given by a transition matrix with representative element π (s s). 8

10 Below, the evolution of the state s will also govern the evolution of the country s bargaining position with creditors. The sovereign country is represented by an agent that maximizes the discounted expected value of its utility from consuming state contingent sequences of the single consumption good {c t (s t )} according to β t π ( ) ( ( s t s )) 0 u ct s t. s t s 0 t=0 Here, the felicity function u is twice continuously differentiable, strictly increasing and strictly concave so that the country is averse to fluctuations in its consumption. The notation s t s 0 is used to denote a history of the state that begins with state s 0, while π (s t s 0 ) is the product of the associated one-period ahead conditional probabilities. The discount factor β lies between zero and one and is assumed to imply a discount rate in excess of the world interest rate. As a result, international borrowing may be motivated by both a desire to smooth consumption, as well as a desire to tilt a country s consumption profile forward in time. A sovereign country that is not in default enters a period with a new value of the state s, and a level of international debt b. It is assumed that b must lie in the set of debt levels, B, which is finite with cardinality N b, and contains both negative and positive elements, as well as the zero element, where negative elements are interpreted as savings by the country. We let V (b, s) denote the value function of a country of that enters the period with debt b and state s, before the country has decided whether or not to default, which is an N e by N b vector of real numbers. The sovereign s first decision is whether or not to default on its debts. If the sovereign defaults, they receive a payoff given by Ṽ D (b, s), which is a N e by N b vector of real numbers, and which will be determined below when we describe the process by which a country in default bargains with its creditors. If we let V R (b, s) denote the value function of a country that enters the period with debt b and state s, after it has decided to repay it s debts, which is an N e by N b vector of real numbers, then the value function V (b, s) satisfies { } V (b, s) = max V R (b, s), Ṽ D (b, s). (1) 9

11 If the sovereign country repays its debts, it must decide how much to consume c and how much debt b B to take into the next period. The value function associated with the repayment of debt, V R, is defined by V R (b, s) = max u (c) + β π (s s) V (b, s ), (2) c,b B s S subject to c q (b, s) b e (s) + b. Here, q (b, s) is a N e N b vector of prices today of a bond that pays one unit tomorrow as long as the country does not default, and that depends on the current state s and total borrowing b. It is determined by competition in international credit markets, which we describe next. International Credit Markets We assume that international credit markets are populated by a large number of risk neutral creditors that behave competitively. The opportunity cost of funds for a creditor is given by the world interest rate r w, which we assume is constant. Competition in the international credit market ensures that creditors expect to earn the world interest rate from their investments in the sovereign borrower s bonds. To understand the determinants of the price of a country s bonds, suppose the country issues a total of b claims, each of which pays one unit tomorrow as long as the country does not default. If a creditor were to buy one unit of the country s bonds at price q (b, s), then competition ensures that they must expect to receive (1 + r w ) q (b, s) on average tomorrow. The actual return they receive has two components. First, with some probability 1 p (b, s) the country is expected to repay-in-full which yields a total of one unit. Second, with probability p (b, s) the country defaults. In this case, the country will commence bargaining with its creditors and the creditor will receive a one-in-b share of any returns from this bargaining process. If we let W (b, s ) be a N e N b vector of the total expected discounted values of any settlement on a default on b bonds in state s tomorrow, viewed from the 10

12 perspective of tomorrow, then the equilibrium bond price must satisfy q (b, s) = 1 p (b, s) + p (b, s) s S π (s s) W (b, s )/b 1 + r w. The total expected discounted value of any settlement, viewed from tomorrow, W (b, s ) will be determined along with the N e N b vector of values to the country from default Ṽ D (b, s), as a result of the bargaining process which we describe in the next section. For now, we assume that W (b, s ) is bounded below by zero and above by b, which in turn ensures that the bond price function takes values in the interval [0, 1/ (1 + r w )] ; we prove that W has these properties below. We let Q (B S) be the set of all functions on B S taking values in [0, 1/ (1 + r w )]. It remains to describe the probability of default p (b, s), which is determined by the sovereign s decision to default described in (1) above. For most values of (b, s), the sovereign country will strictly prefer defaulting over repaying, or repaying over defaulting. However, it is possible that for some values of (b, s) that the country is indifferent. To deal with this possibility, we define an indicator correspondence for default with debt b in state s, Φ (b, s), as Φ (b, s) = 1 if Ṽ D (b, s) > V R (b, s) 0 if Ṽ D (b, s) < V R (b, s) [0, 1] if Ṽ D (b, s) = V R (b, s). From this we can define the default probability correspondence for debt b and state s, P (b, s), as the set of all p (b, s) constructed as p (b, s) = s S φ (b, s ) π (s s), for some φ (b, s) Φ (b, s). Debt Restructuring Negotiations In this subsection, we specify the process by which a sovereign country in default bargains with its creditors over a settlement. We abstract from the coordination problems in debt restructuring negotiations studied by Pitchford and Wright (2007, 2008), and assume 11

13 that creditors are able to perfectly coordinate in bargaining with the country. Hence, our restructuring negotiations are modeled as a game between two players: the sovereign borrower in default, and a single creditor. Environment We assume that the country is in autarky in the period in which the default actually occurs. Hence, the relationship between the total value to creditors from a settlement W (b, s ) and the value to the country from default Ṽ D (b, s), that we introduced above, and the N e N b vectors of outcomes of bargaining that we derive below, W (b, s ) and V D (b, s), is given by W (b, s) = δe [W (b, s ) s ], Ṽ D (b, s) = u ( e def (s) ) + βe [ V D (b, s ) s ]. Here, δ = 1/ (1 + r w ) while e def (s) is used to denote the possibility that the endowment process may be lower in the event of a default (reflecting any direct costs of default). The output loss, combined with one period of autarky, ensure that there is always some cost to default, and deter the country from continually renegotiating its debts. The timeline of actions is described in Figure 3. Negotiations begin with a sovereign country that has previously entered default with a level of debt b. At stake is the ability of the country to re-access credit markets. The value to the country of settling today in state s with its creditors and re-accessing capital markets with a new level of debt b is given by s S π (s s) V (b, s ), where V was described above and is treated as exogenous for the purposes of bargaining. Neither player is able to commit to a split of surplus beyond the current period. Instead, the players can only agree to a current transfer of resources that may be partially (or wholly) financed by the issue of new debt securities. The ability to share future surplus is therefore limited by the fact that the country may default on these new debt securities in the future. Delay can occur as both the creditor and the debtor wait for an improvement in the terms under which new debt securities can be issued. Importantly, the same commitment problem that leads to default also drives the outcome of the renegotiation. 12

14 If no agreement is reached this period, the bargaining game continues with a new state s and the same level of debt b. The assumption that the amount of debt in default, b, is unchanged throughout negotiations captures the fact that for most of the period under study, interest on missed payments was not a part of default settlements 4. Negotiations between the creditors and the debtor are efficient, in the sense that agreements are optimal for the two parties subject to the constraints on negotiations implied by future default risk. To capture this fact, we say that negotiations are privately optimal ex post. Nonetheless, delay may be said to be socially wasteful ex post, as the country is unable to access capital markets while in default, and thus forfeits potential gains from trade in tilting and smoothing its consumption. Timing and Actions Bargaining occurs according to a randomly alternating offer bargaining game with an outside option available to the debtor. The timing is illustrated in Figure 2. At any point, the debtor country has the option of paying off the defaulted debt in full, using any desired mix of current transfers and new debt securities issued at the market price. We refer to this action as the outside option of the debtor, although we stress that this is strictly only an outside option for the game conditional on default, and not for the entire borrowing environment. In addition to being a feature of the actual environment governing sovereign debt renegotiations, this assumption guarantees that the total value of the settlement never exceeds b which serves to bound our bond price function. In every period and in each state of the world s, either the sovereign borrower or the creditor is selected to be the proposer who is then allowed to make a settlement offer. A proposal consists of a transfer of resources τ to the creditor in the current period, and an issue of new debt securities b. The proposer s action is therefore given by an offer of two values (τ, b ) R B. We do not place any additional bounds on the issue of new debt, although debt issues will continue to be limited by the price that new creditors will be prepared to offer for these new bonds. Importantly, we allow for the possibility that the settlement may contain an amount of new money in which case the country receives a positive flow of the 4 In cases that went to court, the courts did not award interest on missed payments until 1997 as part of the legal proceedings involving Elliot Associated and Peru. 13

15 consumption good in the period in which they settle (this corresponds to a negative τ). Once a proposal is made, the non-proposing agent chooses to either accept or reject the current proposal. If the proposal is accepted, or if the debtor country s outside option is taken, the bargaining concludes and the country emerges from default with the new negotiated debt level. If the proposal is rejected and the outside option is not taken, the game continues to the next period, and we say that there has been delay in bargaining. In the next period, the realization of the state determines the identity of the proposer, and the timing repeats with the next proposer suggesting an offer. A history of the bargaining game is a list of all previous actions and states that have occurred after a country s most recent default. That is, we are assuming that each debt restructuring is not affected by previous borrowing, default or debt restructurings, except insofar as these decisions have determined the debt level b. If no offer has been accepted, and if t indexes stages, a history up to the beginning of stage t is defined by the sequence of realizations for the state variable and the sequence of rejected offers: h t = { s t = (s 0, s 1,..., s t 1 ), (τ, b ) t = ( (τ 0, b 0), (τ 1, b 1),..., ( τ t 1, b t 1)) }. We let H t denotes the set of all histories to stage t. Strategies Strategies map the level of the defaulted debt b and the history into a choice of actions. The current state determines the identity of the current proposer, and the set of feasible actions depends on which player is the proposer. A strategy for the creditor when they are the proposer is a function σ C,P : B H t S R B. The situation is more complicated when the debtor is the proposer due to the fact that the debtor may elect to take the outside option. In particular, a strategy for the debtor when they are the proposer is a function σ D,P : B H t S R B {0, 1}, where the third element takes on the value one if the debtor takes the outside option; whether or not the debtor takes the outside option, there is an associated transfer and new debt level (τ, b ). A strategy for the creditor when they are not the proposer depends on whether or not the debtor has taken the outside option. If the debtor has not taken the outside option, a strategy for a non-proposing creditor is a function 14

16 σ C,NP : B H t+1 {0, 1} where 0 denotes rejection of the proposal, and 1 acceptance of the proposal. If the debtor has taken the outside option, the creditor has no choice but to accept the proposed settlement and so a strategy for a non-proposing creditor is a function σ C,NP : B H t+1 {1}. A strategy for the debtor when they are not the proposer is a function σ D,NP : B H t+1 {0} {1} {2} {(τ, b ) R B : τ + q (b, s t+1 ) b b} where the first element 0 indicates a rejection, 1 indicates acceptance, and the third element indicates that the outside option was chosen with associated transfer and new debt levels (τ, b ). A strategy profile is a pair of strategies, one for each player. Payoffs and Equilibrium Next we discuss outcomes and payoffs and define an equilibrium. An outcome is a termination of negotiations plus the final accepted offer. That is, an outcome of the bargaining game is a stopping time t and the associated proposal (τ, b ). At any history, a strategy profile induces an outcome and hence a payoff for each player. The payoff to the debtor given outcome ϕ = {t, (τ, b )} after history s t is V ( D t, s t, (τ, b ) ) t 1 = β r u ( e def (s r ) ) { ( + β t u e def (s t ) τ ) + βe [V (b, s t +1) s t ] }, r=0 while to the creditor it is given by W ( t, s t, (τ, b ) ) = δ t {τ + q (b, s t ) b }. Let G(b, h t ) denote the game from date t onwards starting from history h t. Let h t denote the restriction to the histories consistent with h t. Then σ h t is a strategy profile on G(b, h t ). We let ϕ (σ h t ) be the outcome generated by the strategy profile σ h t in game G (b, h t ). A strategy profile is subgame perfect (SP) if, for every history h t, σ h t is a Nash equilibrium of G(b, h t ). That is W ( ϕ ( σ h t)) W ( ϕ ( σ D h t, σ C h t)), V D ( ϕ ( σ h t)) V D ( ϕ ( σ D h t, σ C h t)), for all σ, t, and h t. 15

17 As is customary in the literature, we impose the restriction of stationarity. A strategy profile is stationary if the actions prescribed at any history depend only on the current state and proposal. That is a stationary strategy profile satisfies: σ D ( b, h t, s t ) = σ D (b, s t ) σ C ( b, ( h t, (s t, (τ t, b t)) )) = σ C (b, s t, (τ t, b t)), for all h t and all t when s t is such that the debtor proposes, and σ C ( b, h t, s t ) = σ C (b, s t ) σ D ( b, ( h t, (s t, (τ t, b t)) )) = σ D (b, s t, (τ t, b t)), for all h t and all t when s t is such that the creditor proposes. A stationary subgame perfect equilibrium (SSP) outcome and payoff are the outcome and payoff generated by an SSP strategy profile. We define a stationary outcome as ((B S) µ, µ),where µ = (τ, b ) and where (B S) µ is the set of debt levels b and states s on which an agreement occurs or the outside option is taken, and where (B S) \ (B S) µ is the disagreement set. 3.B Solution to the Bargaining Model The solution to the overall model involves solving a fixed point problem. First, taking as given the solution to the bargaining problem, we solve for the solution to the debtor countries default problem and update the market price of debt. Second, we take the market price of debt and the debtor s value function from repayment and then use these to solve the bargaining problem. An equilibrium is a fixed point of the composition of the two operators. In this section, we focus on the bargaining model, taking as given the form of the solution to the borrowing problem. Recursive Problem Statement For this section, we take the solution to the borrowing problem as given. That is, the debtor country s value of accessing capital markets V (b, s) is assumed to be a fixed element of the set all real valued N e by N b vectors, and the equilibrium bond price function q (b, s) is 16

18 assumed to be a fixed element of Q (B S). Given these assumptions, we then show that the SSP values of the bargaining game are fixed points of a particular functional equation. As is usual, the key to the approach is that we focus directly on SSP payoffs, rather than on the SSP itself. Our approach is recursive, and relies upon the following operator ˆT. Given any pair of functions (f 1, f 2 ) with f i : B S R for i = 1, 2, we define the mapping ˆT such that: If s is such that the debtor is the proposer ˆT f 1 (b, s) = max max τ,b u(e def (s) τ) + βe[v (b, s ) s], u(e def (s)) + βe [f 1 (b, s ) s] s.t τ + b q(b, s) min {b, δe[f 2 (b, s ) s]}, and ˆT f 2 (b, s) = min {b, δe[f 2 (b, s ) s]}, while if s is such that the creditor is the proposer ˆT f 2 (b, s) = max min b, max τ,b τ + b q(b, s) s.t u(e def (s) τ) + βe[v (b, s ) s] u(e def (s)) + βe [f 1 (b, s ) s], δe [f 2 (b, s ) s], and ˆT f 1 (b, s) = max u(edef (s)) + βe [f 1 (b, s ) s], max τ,b u(e def (s) τ) + βe[v (b, s ) s] s.t τ + b q(b, s) b. Intuitively, the ˆT mapping yields the values from bargaining at a given stage with defaulted debt b and current state s, given that the continuation values associated with not reaching agreement this period are determined by f 1, for the debtor, and f 2 for the creditor. To understand this mapping, note that if the debtor is the proposer, they have three options. First, they could make an offer which will not be accepted. In this case, the debtor consumes the autarky endowment level this period and moves on the next stage with defaulted debt still at b, new state s and payoffs encoded in f 1, while the creditor receives nothing today and a future payoff encoded by f 2. This payoff is the right hand component of the debtor-proposer 17

19 half of the operator, for both the debtor and the creditor. Second, the debtor could take the outside option, in which case the creditor receives the value of the defaulted debt b, and the debtor receives the maximum value achievable while still delivering a payoff of b to the creditor. This corresponds to the left hand side of the creditors part of the debtor-proposer half of the operator, and to the left hand side of the debtor s part of the operator given the constraint on creditor utility defined by b. Third, the debtor could make an offer that is accepted. In this case, since the debtor makes the offer, the creditor receives none of the surplus from the agreement, and hence receives the same payoff as if the offer was not accepted (the right hand side of the creditor part of the debtor-proposer half of the operator). The debtor, on the other hand, receives the maximum value that can be achieved while delivering this value to the creditor (the left hand side of the debtor s part of the operator with the constraint defined by the reservation payoff of the creditor). Since the debtor would never take the outside option when it can do better by making an offer that is accepted, the minimum over the value of the debt and the creditors reservation value is the relevant determinant of the constraint. Similar logic underlies the half of the operator that applies to states in which the creditor is the proposer, noting that the creditor will extract all of the surplus from an accepted proposal up to a maximum value of b at which level the debtor will take the outside option. The following theorem, which can be thought of as a version of the principle of optimality for our problem, establishes an equivalence between SSP payoffs and fixed points of the ˆT operator. Theorem 1. The functions f = (f 1, f 2 ) are SSP payoffs if and only if ˆT f = f. Proof. See Appendix B. This operator forms the basis for our theoretical and numerical analysis of the bargaining problem below. In the next subsection we establish existence of an equilibrium bargain, and provide a sufficient condition under which this bargain is unique, by studying the properties of the ˆT operator. 18

20 Existence and Uniqueness of Symmetric Subgame Perfect Equilibria Next we show that an SSP equilibrium exists, by demonstrating that our ˆT mapping operates on a bounded space of functions, and is monotone. The details are relegated to the appendix. Theorem 2. An SSP equilibrium exists. Proof. See Appendix B. The uniqueness of the values of the equilibrium bargain could be easily established if ˆT is a contraction mapping. However, as in many multi-agent problems, this is not straightforward. The difficulty results from two issues. First, changes in one agent s continuation value function will affect the result of the operator on the other agents continuation value function, because continuation values act as constraints on the proposals that will be accepted. Second, and more importantly, the rates at which changes in one agent s continuation value affect the operator on the other agents continuation value can vary when payoffs are non-linear functions of outcomes. To understand this difficulty, it is instructive to consider how these issues appear in an attempt to establish Blackwell s sufficient conditions for a contraction mapping, and in particular by affecting the proof of the discounting condition. Suppose we change the value of the creditor s and debtor s continuation values by a small constant amount. The discounting property requires that the operator produce functions that are bounded by the modulus of the contraction mapping, which is strictly less than one. Since the country s felicity function is non-linear, it is possible that a small increase in the creditor s continuation value, which would lead to a small change in the settlement value, could lead to a large change in the country s payoff if the marginal utility of consumption was high near the solution of the debtor s problem in the debtor s half of the ˆT operator. Moreover, it is also possible that a small change in the debtor s continuation value could result in a large change in the value of the settlement (and hence also a large change in creditor payoffs) if the marginal utility of consumption is low near the solution of the creditor s problem in the creditor s half of the ˆT operator. 19

21 The following theorem states a condition that is sufficient to prove uniqueness, by imposing bounds on the rate at which resources can be transformed into utility, and the rate at which utility can be transformed into resources. As a consequence of the fact that we have imposed few restrictions on the shape of the V and q functions, the condition is stated in terms of bounds on the slope of the utility function of the debtor. In our numerical work below, as in much of the quantitative literature on sovereign debt and default, we focus on discount factors for the country that are substantially less than one, reflecting political economy problems in developing countries that lead to impatient policy making. For such parameter values, we can typically show that the sufficient condition is satisfied. Theorem 3. Let u : R R be differentiable. If there exists K L > β and K U < 1/δ such that 1/K L u (c) K U, for all c, then the SSP equilibrium values are unique. Proof. See Appendix B. 3.C Solution to Borrowing Model In the previous section, we establish the existence and uniqueness of a solution to the debt restructuring bargaining problem, taking as given the value to the country from re-accessing capital markets with new debt b, E [V (b, s ) s], and the value of new debt to creditors q (b, s). In this section, we take as given the solution to the bargaining model, and hence the value to the country and the creditor from being in default, and then establish existence of a solution to the borrowing problem. That is, we take as given the N e N b vectors of payoffs to the country, Ṽ D (b, s), and the creditor, W (b, s), in default, that are elements of B (B S). The solution of the borrowing problem is established as the composition of two operators. The first takes a value to the country from default and an equilibrium bond price function, and then solves the country s problem to obtain a value to the country for access to capital markets, and a default policy function, which is a selection from a default policy correspondence. The second takes the default policy function and combines it with the value to the creditors from default to obtain a new bond price function. Existence of a solution follows from the monotonicity of the composition of these operators. 20

22 Theorem 4. Given (Ṽ D (b, s), W (b, s)) B (B S) and q (b, s) Q (B S), there exists a value function for the country, V (b, s), and an equilibrium bond price function q (b, s) Q (B S), that solve the borrowing problem. Proof. See Appendix B. Given the result of this Theorem, it is tempting to try to prove existence of an equilibrium for our entire model by iterating successively on the T V, T q and ˆT operators. However, this approach need not converge. Specifically, although iterating on the T V and T q operators produces a monotone operator, when combined with the bargaining operator, the compounded operator need not be monotone. Intuitively, it can be the case that a high value to the creditor in default, and a low value to debtor, leads to a high bond price, which in turn leads to a high value to the country from repayment. This high value to repayment can lead to a high value from default, which then leads to a low bond price in the next iteration. That is, we cannot rule out cycles in the successive application of these operators. In the next section, we describe an alternative method for proving existence. 3.D Existence of Equilibrium In this section, we establish the existence of a recursive equilibrium for our economy. First, we define an equilibrium for our economy. Definition 1. An equilibrium for our economy is a value function for the country from borrowing V (b, s), a value function for the country in default V D (b, s), a value function for the creditor in default W (b, s) and a bond price function q (b, s) such that: 1. Given the bond price function q (b, s) and the value to the country from re-accessing capital markets V (b, s), the country and the creditor optimally bargain over re-access to financial markets. ˆT ; That is, V D and W are fixed points of the inside default operator 2. Given the value to the country and from default V D (b, s), and the bond price q (b, s), the country makes optimal borrowing and default decisions. That is, V (b, s) is a fixed point of T V with associated default policy correspondence Φ (b, s) 21

23 3. Given the payoff to the creditor in default W and the optimal default policy correspondence, the bond price function q (b, s) satisfies the no arbitrage condition for creditors. That is, q (b, s) is a fixed point of the operator T q. The latter two conditions may equivalently be written as: Given V D (b, s) and W (b, s), V (b, s) and q (b, s) are a fixed point of the outside default operator, which is the composition of the T V and T q operators. We prove existence by using the operators defined above to construct a new mapping from the space of value functions for the country and creditor in default, and the space of bond price functions, into itself, and establishing that it possesses a fixed point. Specifically, define the mapping H from B (B S) Q (B S) into itself as follows. First, given V D, W and q, iterate on the outside default operator to convergence to obtain a new bond price function q (b, s). Second, given V D and q, iterate on the T V operator to convergence to produce a value function V. Then, given the old q and this V, iterate on the ˆT operator to convergence to find new V D and W. We establish that the combination of these operators defines an upper hemi-continuous correspondence with non-empty and convex values. Then, noting that B (B S) Q (B S) is a compact and convex space of functions, the result then follows by application of the Kakutani-Fan-Glicksberg fixed point theorem. Theorem 5. There exists an equilibrium of our borrowing economy. Proof. See Appendix B. 4 Calibration and Numerical Results In this section, we present results from several numerical solutions of the model that vary only in the calibration of the bargaining power process for the debtor and creditor. These examples are used to illustrate some comparative static properties of the model, and to build intuition for the elements of the model that produce delay. We then present our benchmark case in which the parameters of the model governing bargaining power are calibrated to the relationship between default and output observed in the data. The model is then assessed according to it s ability to match the other facts discussed in the introduction. 22